GPU Computing in Julia

Biostat/Biomath M257

Author

Dr. Hua Zhou @ UCLA

Published

April 17, 2025

This lecture introduces GPU computing in Julia.

1 GPGPU

GPUs are ubiquitous in modern computers. Following are NVIDIA GPUs on today’s typical computer systems.

NVIDIA GPUs H100 PCIe RTX 6000 RTX 5000
H100 RTX 6000 RTX 5000
Computers servers, cluster desktop laptop
Server Desktop Laptop
Main usage scientific computing daily work, gaming daily work
Memory 80 GB 48 GB 16 GB
Memory bandwidth 2 TB/sec 960 GB/sec 576 GB/sec
Number of cores ??? ??? ???
Processor clock ??? GHz ??? GHz ??? GHz
Peak DP performance 26 TFLOPS ??? TFLOPS ??? TFLOPS
Peak SP performance 51 TFLOPS 91.1 TFLOPS 42.6 TFLOPS

2 GPU architecture vs CPU architecture

  • GPUs contain 1000s of processing cores on a single card; several cards can fit in a desktop PC

  • Each core carries out the same operations in parallel on different input data – single program, multiple data (SPMD) paradigm

  • Extremely high arithmetic intensity if one can transfer the data onto and results off of the processors quickly

i7 die Fermi die
Einstein Rain man

3 GPGPU in Julia

GPU support by Julia is under active development. Check JuliaGPU for currently available packages.

There are multiple paradigms to program GPU in Julia, depending on the specific hardware.

  • CUDA is an ecosystem exclusively for Nvidia GPUs. There are extensive CUDA libraries for scientific computing: CuBLAS, CuRAND, CuSparse, CuSolve, CuDNN, …

    The CUDA.jl package allows defining arrays on Nvidia GPUs and overloads many common operations.

  • The AMDGPU.jl package allows defining arrays on AMD GPUs and overloads many common operations.

  • The Metal.jl package allows defining arrays on Apple Silicon and overloads many common operations.

    AppleAccelerate.jl wraps the macOS Accelerate framework, which provides high-performance libraries for linear algebra, signal processing, and image processing on Apple Silicon.

  • The oneAPI.jl package allows defining arrays on Intel GPUs and overloads many common operations.

I’ll illustrate using Metal.jl on my MacBook Pro running MacOS Sequoia 15.4. It has Apple M2 chip with 38 GPU cores.

versioninfo()
Julia Version 1.11.5
Commit 760b2e5b739 (2025-04-14 06:53 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (arm64-apple-darwin24.0.0)
  CPU: 12 × Apple M2 Max
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, apple-m2)
Threads: 8 default, 0 interactive, 4 GC (on 8 virtual cores)
Environment:
  JULIA_NUM_THREADS = 8
  JULIA_EDITOR = code

Load packages:

using Pkg

Pkg.activate(pwd())
Pkg.instantiate()
Pkg.status()
  Activating project at `~/Documents/github.com/ucla-biostat-257/2025spring/slides/09-juliagpu`
Status `~/Documents/github.com/ucla-biostat-257/2025spring/slides/09-juliagpu/Project.toml`
  [13e28ba4] AppleAccelerate v0.4.0
  [6e4b80f9] BenchmarkTools v1.6.0
  [bdcacae8] LoopVectorization v0.12.172
  [dde4c033] Metal v1.5.1
  [37e2e46d] LinearAlgebra v1.11.0

4 Query GPU devices in the system

using Metal, AppleAccelerate

Metal.versioninfo()
macOS 15.4.0, Darwin 24.4.0

Toolchain:
- Julia: 1.11.5
- LLVM: 16.0.6

Julia packages: 
- Metal.jl: 1.5.1
- GPUArrays: 11.2.2
- GPUCompiler: 1.3.2
- KernelAbstractions: 0.9.34
- ObjectiveC: 3.4.1
- LLVM: 9.2.0
- LLVMDowngrader_jll: 0.6.0+0

1 device:
- Apple M2 Max (4.000 GiB allocated)

5 Transfer data between main memory and GPU

using Random
Random.seed!(257)

# generate SP data on CPU
x = rand(Float32, 3, 3)
# transfer data form CPU to GPU
xd = MtlArray(x)
3×3 MtlMatrix{Float32, Metal.PrivateStorage}:
 0.145793  0.939801  0.479926
 0.567772  0.577251  0.81655
 0.800538  0.38893   0.914135
# generate array on GPU directly
# yd = Metal.ones(3, 3)
yd = MtlArray(ones(Float32, 3, 3))
3×3 MtlMatrix{Float32, Metal.PrivateStorage}:
 1.0  1.0  1.0
 1.0  1.0  1.0
 1.0  1.0  1.0
# collect data from GPU to CPU
x = collect(xd)
3×3 Matrix{Float32}:
 0.145793  0.939801  0.479926
 0.567772  0.577251  0.81655
 0.800538  0.38893   0.914135

6 Linear algebra

using BenchmarkTools, LinearAlgebra, Random

Random.seed!(257)

n = 2^14
# on CPU
x = rand(Float32, n, n)
y = rand(Float32, n, n)
z = zeros(Float32, n, n)
# on GPU
xd = MtlArray(x)
yd = MtlArray(y)
zd = MtlArray(z);

6.1 Dot product

# SP matrix dot product on GPU: tr(X'Y)
# why are there allocations?
bm_gpu = @benchmark Metal.@sync dot($xd, $yd)
BenchmarkTools.Trial: 659 samples with 1 evaluation per sample.
 Range (minmax):  7.475 ms 8.975 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     7.576 ms               GC (median):    0.00%
 Time  (mean ± σ):   7.591 ms ± 99.875 μs   GC (mean ± σ):  0.00% ± 0.00%
        ▂▄█▆▇▅▅▂▂▅▂▂▁ ▂▄▂                                   
  ▄▆▄▆███████████████▇████▇▇▆▇▄▅▄▃▄▃▃▄▂▂▃▂▂▄▂▁▁▁▃▁▁▂▁▁▂▁▁▂ ▄
  7.47 ms        Histogram: frequency by time        7.85 ms <
 Memory estimate: 21.24 KiB, allocs estimate: 837.
# SP matrix dot product on CPU: tr(X'Y)
bm_cpu = @benchmark dot($x, $y)
BenchmarkTools.Trial: 141 samples with 1 evaluation per sample.
 Range (minmax):  34.021 ms40.975 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     34.774 ms               GC (median):    0.00%
 Time  (mean ± σ):   35.469 ms ±  1.405 ms   GC (mean ± σ):  0.00% ± 0.00%
    ▅█▂                                                        
  ▄▅███▇▄▅▃▅▁▃▅▁▅▃▃▅▄▁▁▃▄▃▄▃▃▅▆▄▅▁▁▁▁▃▁▁▁▁▁▁▁▁▃▁▁▃▁▁▁▁▁▁▁▁▃ ▃
  34 ms           Histogram: frequency by time        40.6 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.
# speedup
median(bm_cpu.times) / median(bm_gpu.times)
4.590047354266009

6.2 Broadcast

# SP broadcast on GPU: z .= x .* y
# why is there allocation?
bm_gpu = @benchmark Metal.@sync $zd .= $xd .* $yd
BenchmarkTools.Trial: 547 samples with 1 evaluation per sample.
 Range (minmax):  8.761 ms 11.350 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     9.051 ms                GC (median):    0.00%
 Time  (mean ± σ):   9.135 ms ± 347.959 μs   GC (mean ± σ):  0.00% ± 0.00%
    ▄▇█▇█▁▁▁▅▂                                               
  ▃▆██████████▇▆▆▆▇▄▄▄▃▃▃▃▃▁▃▄▃▁▂▃▁▃▂▂▂▁▃▁▃▁▂▃▁▂▃▁▂▁▁▁▂▁▂▁▂ ▄
  8.76 ms         Histogram: frequency by time        10.6 ms <
 Memory estimate: 4.53 KiB, allocs estimate: 177.
# SP broadcast on CPU: z .= x .* y
bm_cpu = @benchmark $z .= $x .* $y
BenchmarkTools.Trial: 139 samples with 1 evaluation per sample.
 Range (minmax):  34.889 ms45.114 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     35.275 ms               GC (median):    0.00%
 Time  (mean ± σ):   35.983 ms ±  1.944 ms   GC (mean ± σ):  0.00% ± 0.00%
  ▇█▄▂▁  ▁                                                    
  █████▆█▁▁▅█▅▁▁▅▁█▁▅▁▅▅▁▁▁▁▁▅▁▁▁▁▁▅▁▁▁▁▅▅▁▅▁▁▁▁▁▅▅▁▁▁▁▁▁▁▅ ▅
  34.9 ms      Histogram: log(frequency) by time      44.8 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.
# speedup
median(bm_cpu.times) / median(bm_gpu.times)
3.8975395957005157

6.3 Matrix multiplication

# SP matrix multiplication on GPU
bm_gpu = @benchmark Metal.@sync mul!($zd, $xd, $yd)
BenchmarkTools.Trial: 6 samples with 1 evaluation per sample.
 Range (minmax):  918.484 ms  1.044 s   GC (min … max): 0.00% … 0.00%
 Time  (median):     983.040 ms               GC (median):    0.00%
 Time  (mean ± σ):   978.952 ms ± 48.398 ms   GC (mean ± σ):  0.00% ± 0.00%
  █      █            █                    █ █               █  
  █▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
  918 ms          Histogram: frequency by time          1.04 s <
 Memory estimate: 960 bytes, allocs estimate: 54.

For this problem size on this machine, we see GPU achieves a staggering 9 TFLOPS throughput with single precision!

# SP throughput on GPU
(2n^3) / (minimum(bm_gpu.times) / 1e9)
9.576750193921969e12
# SP matrix multiplication on CPU
bm_cpu = @benchmark mul!($z, $x, $y)
BenchmarkTools.Trial: 2 samples with 1 evaluation per sample.
 Range (minmax):  3.960 s  4.020 s   GC (min … max): 0.00% … 0.00%
 Time  (median):     3.990 s               GC (median):    0.00%
 Time  (mean ± σ):   3.990 s ± 42.588 ms   GC (mean ± σ):  0.00% ± 0.00%
                              ▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
  3.96 s         Histogram: frequency by time        4.02 s <
 Memory estimate: 0 bytes, allocs estimate: 0.
# SP throughput on CPU
(2n^3) / (minimum(bm_cpu.times) / 1e9)
2.221173865839729e12

We see >10x speedup by GPUs in this matrix multiplication example.

# cholesky on Gram matrix
# This one doesn't seem to work on Apple M2 chip yet
xtxd = xd'xd + I
@benchmark Metal.@sync cholesky($(xtxd))
ArgumentError: ArgumentError: cannot take the CPU address of a MtlMatrix{Float32, Metal.PrivateStorage}
ArgumentError: cannot take the CPU address of a MtlMatrix{Float32, Metal.PrivateStorage}



Stacktrace:

  [1] unsafe_convert(::Type{Ptr{Float32}}, x::MtlMatrix{Float32, Metal.PrivateStorage})

    @ Metal ~/.julia/packages/Metal/N2ABH/src/array.jl:266

  [2] potrf!(uplo::Char, A::MtlMatrix{Float32, Metal.PrivateStorage})

    @ LinearAlgebra.LAPACK ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/lapack.jl:3293

  [3] _chol!

    @ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:187 [inlined]

  [4] #cholesky!#163

    @ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:268 [inlined]

  [5] cholesky!

    @ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:267 [inlined]

  [6] cholesky!(A::MtlMatrix{Float32, Metal.PrivateStorage}, ::NoPivot; check::Bool)

    @ LinearAlgebra ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:301

  [7] cholesky! (repeats 2 times)

    @ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:295 [inlined]

  [8] _cholesky

    @ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:411 [inlined]

  [9] cholesky (repeats 2 times)

    @ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:401 [inlined]

 [10] macro expansion

    @ ~/.julia/packages/Metal/N2ABH/src/utilities.jl:10 [inlined]

 [11] var"##core#282"(xtxd#281::MtlMatrix{Float32, Metal.PrivateStorage})

    @ Main ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:598

 [12] var"##sample#283"(::Tuple{MtlMatrix{Float32, Metal.PrivateStorage}}, __params::BenchmarkTools.Parameters)

    @ Main ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:607

 [13] _lineartrial(b::BenchmarkTools.Benchmark, p::BenchmarkTools.Parameters; maxevals::Int64, kwargs::@Kwargs{})

    @ BenchmarkTools ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:186

 [14] _lineartrial(b::BenchmarkTools.Benchmark, p::BenchmarkTools.Parameters)

    @ BenchmarkTools ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:181

 [15] #invokelatest#2

    @ ./essentials.jl:1055 [inlined]

 [16] invokelatest

    @ ./essentials.jl:1052 [inlined]

 [17] #lineartrial#46

    @ ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:51 [inlined]

 [18] lineartrial

    @ ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:50 [inlined]

 [19] tune!(b::BenchmarkTools.Benchmark, p::BenchmarkTools.Parameters; progressid::Nothing, nleaves::Float64, ndone::Float64, verbose::Bool, pad::String, kwargs::@Kwargs{})

    @ BenchmarkTools ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:299

 [20] tune!

    @ ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:288 [inlined]

 [21] tune!(b::BenchmarkTools.Benchmark)

    @ BenchmarkTools ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:288

 [22] top-level scope

    @ ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:461
xtx = collect(xtxd)
@benchmark LinearAlgebra.cholesky($(Symmetric(xtx)))
BenchmarkTools.Trial: 1 sample with 1 evaluation per sample.
 Single result which took 7.707 s (0.00% GC) to evaluate,
 with a memory estimate of 1.00 GiB, over 3 allocations.
@benchmark AppleAccelerate.cholesky($(Symmetric(xtx)))
BenchmarkTools.Trial: 1 sample with 1 evaluation per sample.
 Single result which took 7.550 s (0.01% GC) to evaluate,
 with a memory estimate of 1.00 GiB, over 3 allocations.

We don’t see GPU speedup of Cholesky at the moment.

7 Evaluation of elementary and special functions on GPU

# elementwise function on GPU arrays
fill!(yd, 1)
bm_gpu = @benchmark Metal.@sync $zd .= log.($yd .+ sin.($xd))
bm_gpu
BenchmarkTools.Trial: 562 samples with 1 evaluation per sample.
 Range (minmax):  8.789 ms 9.483 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     8.879 ms               GC (median):    0.00%
 Time  (mean ± σ):   8.896 ms ± 82.151 μs   GC (mean ± σ):  0.00% ± 0.00%
   ▁ ▂▂▂█▃▄▆▅▂▇ ▂▂▁▃                                        
  ▄████████████████▇▆▆▆▄▅▅▃▄▄▇▅▄▄▂▄▃▂▁▂▂▂▂▂▁▂▁▂▃▁▁▂▁▂▁▁▁▂ ▄
  8.79 ms        Histogram: frequency by time         9.2 ms <
 Memory estimate: 4.53 KiB, allocs estimate: 177.
# elementwise function on CPU arrays
x, y, z = collect(xd), collect(yd), collect(zd)
bm_cpu = @benchmark $z .= log.($y .+ sin.($x))
bm_cpu
BenchmarkTools.Trial: 2 samples with 1 evaluation per sample.
 Range (minmax):  2.721 s  2.749 s   GC (min … max): 0.00% … 0.00%
 Time  (median):     2.735 s               GC (median):    0.00%
 Time  (mean ± σ):   2.735 s ± 19.800 ms   GC (mean ± σ):  0.00% ± 0.00%
                              ▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
  2.72 s         Histogram: frequency by time        2.75 s <
 Memory estimate: 0 bytes, allocs estimate: 0.
# Speed up
median(bm_cpu.times) / median(bm_gpu.times)
307.9922734440778

GPU brings great speedup (>50x) to the massive evaluation of elementary math functions.